Problem: Evaluate the following expression. Your answer must be exact. $\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i\right)^{12}=$
Solution: The Strategy The easiest way to find $z^{n}$ for a complex number $z=({a}+{b}i)$ is using its modulus and argument. Therefore, our solution will consist of the following steps: Find the modulus and argument of $z$. [How is this done, in general?] Find the modulus and argument of $z^{n}$. [How is this done, in general?] Find the rectangular form $z^{n}$. Find the modulus and argument of $\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i\right)$ $\left({-\dfrac{\sqrt{3}}{2}}{-\dfrac{1}{2}}i\right)$ is of the form $({a}+{b}i)$, where ${a=-\dfrac{\sqrt{3}}{2}}$ and ${b=-\dfrac{1}{2}}$. Therefore: $\begin{aligned}r&=\sqrt{{a}^2 + {b}^2} \\\\&=\sqrt{ \left({-\dfrac{\sqrt{3}}{2}}\right)^2 + \left({-\dfrac{1}{2}}\right)^2} \\\\&=\sqrt{{\dfrac{3}{4}}+{\dfrac{1}{4}}} \\\\&=1\end{aligned}$ Using the arctangent formula, we have: $\begin{aligned}\theta&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{-\dfrac{1}{2}}}{{-\dfrac{\sqrt{3}}{2}}}\right) \\\\&=30^\circ\end{aligned}$ Since both ${a=-\dfrac{\sqrt{3}}{2}}$ and ${b=-\dfrac{1}{2}}$ are negative, $\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i\right)$ lies in Quadrant $3$. Therefore, $\theta$ must be between $180^\circ$ and $270^\circ$. Using the identity $\tan(180+\theta)=\tan(\theta)$, we know that the following is also a solution of the equation. $180^\circ+(30^\circ)=210^\circ$ So $\theta = 210^{\circ}$. Find the modulus and argument of $\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i\right)^{12}$ We found that the modulus and argument of $\left({-\dfrac{\sqrt{3}}{2}}{-\dfrac{1}{2}}i\right)$ are $1$ and $210^\circ$. Therefore, the modulus and argument of $\left({-\dfrac{\sqrt{3}}{2}}{-\dfrac{1}{2}}i\right)^{12}$ are $1^{12}=1$ and $(210^\circ)\cdot12=2520^\circ=360^\circ$. Find the rectangular form of $\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i\right)^{12}$ Since the argument is $360°$, we know the number lies on the positive side of the real number axis and is therefore a positive real number. Since the modulus is $1$, our solution is $1$. [What does this look like graphically?] [How do we find this algebraically?] Summary $\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i\right)^{12}=1$